Problem 50 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{x-2}{4}+\frac{x+4}{8}$$

Step-by-Step Solution

Answer

The result is \(\frac{3x}{8}\) in simplest form.

Step 1: Find a common denominator

The denominators of the fractions are 4 and 8. To add these fractions, we need a common denominator. The least common multiple (LCM) of 4 and 8 is 8.

Step 2: Rewrite the first fraction

Convert the first fraction, \(\frac{x-2}{4}\), to have the denominator 8 by multiplying both the numerator and the denominator by 2. This gives us \(\frac{2(x-2)}{8}\).

Step 3: Simplify the rewritten fraction

Simplify \(\frac{2(x-2)}{8}\) to \(\frac{2x-4}{8}\) by distributing the 2 in the numerator.

Step 4: Add the fractions

Now that both fractions, \(\frac{2x-4}{8}\) and \(\frac{x+4}{8}\), have the same denominator, add the numerators: \((2x-4) + (x+4)\).

Step 5: Simplify the expression

Combine like terms in the numerator: \(2x + x - 4 + 4 = 3x\). So, the expression becomes \(\frac{3x}{8}\).

Step 6: Verify simplest form

Ensure that \(\frac{3x}{8}\) is in its simplest form. Since 3 and 8 have no common factors other than 1, the expression is already in simplest form.

Key Concepts

Common Denominator

In order to add or subtract fractions effectively, it's crucial to find a common denominator. A common denominator is a shared multiple of the denominators in two or more fractions, which allows fractions to be easily combined. In our example, we are adding \(\frac{x-2}{4}\) and \(\frac{x+4}{8}\). The denominators here are 4 and 8.
To find a common denominator, we need to determine the least common multiple (LCM) of these numbers. The LCM is the smallest number that both denominators divide into without leaving a remainder.

• The multiples of 4 are: 4, 8, 12, 16, 20,...
• The multiples of 8 are: 8, 16, 24, 32,...

As we can see, the smallest common multiple is 8.
Once we've found this, we can convert the fractions to have this as their new denominator, making them compatible for addition.

Simplifying Fractions

Simplifying fractions is the process of making a fraction as simple as possible. When a fraction is simplified, the numerator and denominator have no common divisors other than 1. This helps in understanding and using the fraction more conveniently. In our example, after finding the common denominator (8), we converted the fraction \(\frac{x-2}{4}\) to \(\frac{2(x-2)}{8}\).
To simplify this, distribute the 2 in the numerator:

• \(2(x-2) = 2x - 4\) leads to \(\frac{2x-4}{8}\)

Once we add this to the other fraction, the combined fraction is \(\frac{3x}{8}\). Here, 3 and 8 have no common factors besides 1, meaning it is already simplified. Simplifying is essential to ensure results are clear and correct.

Like Terms

Combining like terms is a key step in simplifying algebraic expressions. Like terms are terms in an expression that have identical variable parts; this makes them comparable and combinable.
In the fraction addition \((2x-4) + (x+4)\), the terms with the variable 'x' (\(2x\) and \(x\)) are like terms. This means they can be added together:

• \(2x + x = 3x\)

The constants \(-4\) and \(+4\) are also like terms. Here, they conveniently cancel each other out because \(-4 + 4 = 0\).
Recognizing and combining like terms helps in simplifying expressions to their most condensed form, making it easier to work with them in further calculations.